How many dimensions are there in the universe? Could any "extra" dimensions represent the Standard Model? An underlying principle is needed to answer these and similar questions. We believe the key is to reformulate existing theory in terms of two special mathematical structures: the octonions -- the largest division algebra, and the exceptional Jordan algebra -- the largest reasonable matrix algebra over the octonions. For 20 years, we have been carefully examining how to rewrite certain pieces of fundamental physical theories in terms of these structures. It turns out that this can be done only in special cases. For example, Lorentz transformations exist in any dimension, but only in ten dimensions do they fit naturally inside the exceptional Jordan algebra. Every time we have used the octonions to guide our choices, we have discovered new features of these special cases. Instead of being inputs to the theory, motivated by experiment, these facts emerge naturally as consequences of the special properties of these mathematical structures. We plan to explore the tantalizing evidence that these mathematical structures can be used to give a unified theory of fundamental particles and their interactions.

In previous work, we showed how to rewrite the massless Dirac equation in ten dimensions in terms of an octonionic generalization of the two-component Penrose spinors of relativity. The resulting description has several remarkable properties:

An underlying complex structure naturally selects four preferred space-time dimensions, leading to an interpretation in terms of both massive and massless particles;

Each solution can be identified with a classical solution of the four-dimensional Dirac equation, with the correct number of spin states for precisely three generations of fermions, with single-helicity, massless neutrinos.

Our earlier work can be expressed in terms of the ten-dimensional Lorentz group in the form S L(2, O), acting separately on vectors and spinors. This formalism admits a natural extension to an action of E6 ≈ S L(3, O) on the exceptional Jordan algebra that incorporates vectors and spinors in a single framework. We propose here to first extend our two-component description of fermions to a three-component description of all the (free) fundamental particles, then attempt to describe particle interactions in this language.